Pam  00:00

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  00:08

And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:16

I said, Pam. (unclear) Kim Montague.

Kim  00:18

(unclear). I was not listening because I was scrolling to find the intro.

Pam  00:22

To find what you were supposed to say. I love it. Well, well done. You made it there. That was good. We know that algorithms are amazing human achievements, but they're not good teaching tools because, ya'll, mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. 

Kim  00:38

In this podcast, we help you teach, mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:45

We invite you to join us to make math more figure-out-able.

Kim  00:49

You want to know what I was thinking about while I was scrolling?

Pam  00:52

I do.

Kim  00:52

What the question that I asked you right before we got started. So... 

Pam  00:56

I don't remember.

Kim  00:58

I said, "Hey, what math are we going to do today? What am I going to be doing?" And you said, "You're going to a cafe." And I was like, "That's not helpful. Okay. I have no idea what we're doing today."

Pam  01:09

Alright, yeah. So, Kim's going to be a little bit surprised maybe (unclear).

Kim  01:13

Talk about vulnerable, people. 

Pam  01:15

Bam. 

Kim  01:15

I don't even know what we're doing.

Pam  01:17

So much fun. Well, pretty soon you're going to do it to me, so that all will be fair.

Kim  01:21

Okay.

Pam  01:22

Okay, so a few weeks ago. Nira asked for some stuff with systems of equations. And I know we've taken our time, but I did kind of want to build some things with equations and equalities.

Kim  01:32

Mmhm.

Pam  01:32

Today, Kim, we are going to dive in and do maybe the coolest thing I've ever seen with systems of equations. And to be clear, I've seen some cool things. 

Kim  01:41

Okay.

Pam  01:42

I'm totally going to give Kara Imm. a lot of credit. So, Kara was my co-author with the Algebra Problem Strings book. I had a super good time writing with her. She is... I think maybe you and I would call her one of the best Problem String facilitators.

Kim  01:57

Oh, yeah. I got to see her in New York. (unclear).

Pam  02:00

(unclear). I think you saw her twice maybe because you saw her. You saw her and then we saw her together. And...

Kim  02:05

Yeah.

Pam  02:05

...when we saw her together, we were like over on the side going, "Oh my gosh!" and we were comparing notes about things that she did and said. Yeah. So, thanks, Kara for that. Okay, so let's dive in to... Let's go to a cafe. 

Kim  02:17

Okie doke.

Pam  02:17

So, Kim, you go to a cafe and you bought... You're buying for the crew. 

Kim  02:23

Okay.

Pam  02:23

You bought 5 sandwiches and 4 cookies. I don't know who didn't get a cookie. It's not very nice of you, but.  5 sandwiches and 4.

Kim  02:30

4 cookies. 

Pam  02:31

Yeah, 5 sandwiches and (unclear).

Kim  02:32

I only need four cookies. 

Pam  02:33

Oh. For 5 sandwiches. Somebody eats (unclear). 

Kim  02:36

Oh. I have boys in my house. That's... More sandwiches in an option. 

Pam  02:40

Oh, gotcha.

Kim  02:40

(unclear).

Pam  02:40

Okay, alright. Or maybe it's me and I'm on a diet because that's always the thing. 

Both Pam and Kim  02:45

Okay.

Pam  02:45

So, 5 sandwiches and 4 cookies. And you looked at the receipt, and those 5 sandwiches and 4 cookies cost $62.00.

Kim  02:53

Mmm.

Pam  02:54

Before you get going, going, going.

Kim  02:56

Mmhm.

Pam  02:57

Listeners, you might want to pause. So, Kim, don't say anything quite yet. Could you think of a possible price for the sandwiches and cookies that would get you $62.00? So, don't say anything too fast, but... I mean, I guess they can pause. But I'm just going to, you know like, just let you brainstorm.

Kim  03:21

Am I brainstorming? 

Pam  03:22

Yeah. Can you think of a possible...  Yeah, I'll let you talk. Yeah. 

Kim  03:25

Okay, sorry. So, if the sandwiches were $10.00 bucks.

Pam  03:32

Okay.

Kim  03:33

Then, that would be $50.00 of my dollars. So, then that means the 4 cookies would cost $12.00, and so they would be $3.00 each. 

Pam  03:42

Oh, nice. 

Kim  03:43

So, it's $10.00 sandwich, and $3.00 cookie.

Pam  03:45

Nice, and that makes the total lots of sense.

Kim  03:48

Mmhm.

Pam  03:49

Do one more, just for grins. What if the sandwiches were like a $1.00

Kim  03:52

Oh, I put $2.00.

Pam  03:54

Okay, or that. Or that.

Kim  03:55

So, if they were really bad, $2.00 sandwiches, then that would be $10.00. 

Pam  04:00

Okay.

Kim  04:01

Out of my $62.00 bucks, so then $52 for cookies.

Pam  04:06

Oof!

Kim  04:06

Holy cow! So, 4 cookies for $52.00, that would be $13.00 bucks for a cookie (unclear).

Pam  04:12

I so want to know how you just did that, but we're not going to go there today. 

Kim  04:15

Okay. 

Pam  04:16

Okay. Cool. Nice. So, we could continue to do that, right?   (unclear) like. In fact, one way that sometimes I'll do with kids. They're like, "Nah, I don't do numbers." I'll just say, "Well, what if the sandwiches were free?" 

Kim  04:21

Yeah, yeah. (unclear).

Pam  04:21

Or what if the cookies were free? Yeah, just get them started, kind of thinking about some possibilities. We could brainstorm some possibilities, kind of get a sense for that. And then we could kind of ask ourselves like, "How many possibilities are there?" And as we talk about how many possibilities there are, we're going to use technology today to kind of look at all those possibilities. Now, Kim is a self professed not...

Kim  04:21

Yeah, yeah. Not techie.

Pam  04:54

Not... 

Kim  04:54

No tech. 

Pam  04:55

Okay.

Kim  04:56

I'll say that.

Pam  04:57

So, we're just going to like... It will work. So, we're in a Desmos, online graphing calculator. 

Kim  05:03

Yep.

Pam  05:03

Neither of us can see each other, so we're going to (unclear)...

Kim  05:05

Uh-oh, okay. (unclear).

Pam  05:07

...what we're saying. Kim, if we were to just type in kind of what we just said which was 5 times the cost of the sandwiches. So, I'm going to put 5x. And x is going to represent the cost of the sandwiches.

Kim  05:19

Okay.

Pam  05:20

Plus. Because we're going to add it to the cost of the cookies.

Kim  05:22

Yep.

Pam  05:23

Plus 4 times y. And we'll let y be the cost of the cookies. Now, Desmos users, we could have put s and c. We could have.

Kim  05:31

Oh, I was going to ask about that. Okay.

Pam  05:33

We could have. I don't know why I'm not today. Today, I'm not. 

Kim  05:36

It's fine. 

Pam  05:36

And then when we sum those up on the bill, it was $62.00 Is that right?

Kim  05:40

Mmhm.

Pam  05:41

Yeah? So, I just typed equals $62.00 What do you see, Kim. So, 5x plus 4y equals 62.

Kim  05:46

I see a portion of a line. 

Pam  05:49

Yeah.

Kim  05:50

Yeah. 

Pam  05:51

So, what... 

Kim  05:53

My screen goes from -10 to 10.

Pam  05:57

And about -9 to... Yeah, about -10 to 10, yeah. That's the home screen usually.

Kim  06:03

Yeah, can I... You going to let me... 

Pam  06:05

I am going to. Yes.

Kim  06:05

...find more. Okay.

Pam  06:06

Yeah, yep. What...

Kim  06:08

I'm going to zoom out because I want to see more of the line. And I kind of am interested where they cross the axis. 

Pam  06:15

So, you have this gut instinct.

Kim  06:16

Yeah.

Pam  06:17

That there's something kind of cool about where they cross the x-axis. Okay. Can you just hover over like where it crosses the x-axis? What is that point?

Kim  06:26

I have (12.4, 0)

Pam  06:30

And any idea what that means in this cookie, sandwich? Yeah, what does it mean. 

Kim  06:34

If I bought 12.4 cookie...sandwiches? Oh, I need the s. 12. 12. It would be $12.40 for a sandwich and 0 cost for the cookie.

Pam  06:46

Hey, there's the free cookie. 

Kim  06:48

Yeah.

Pam  06:48

There's the free cookie. So, listeners, you think about while Kim goes and looks for it. What about the y-intercept? Like, where it's 0. And then I'm seeing 15.5. What does that represent in this sandwich, cookie scenario?

Kim  07:04

Tell me the numbers you got. I think I lost it. Oh, here we go. Oh, man.

Pam  07:12

You're so stinkin' fun.

Kim  07:13

If I pay $0.00 for sandwiches, then my cookies are $15.50. That's never happening for me.

Pam  07:19

Bam. That's never... You would not do that?

Kim  07:21

(unclear) for a cookie. That's crazy.  was (unclear).  I mean, if it Maybe it's a cookie cake! It's a cookie cake.

Pam  07:25

Ooh, a cookie cake. Okay.

Both Pam and Kim  07:26

Alright.

Pam  07:26

I was going to say or really good chips and salsa. You would totally do that. 

Kim  07:29

Oh, for sure I would. 

Pam  07:30

Yeah, okay. 

Kim  07:32

Cool. 

Pam  07:32

So, I wonder if we could find one of your... You had said $10.00 for a sandwich and $3.00 for a cookie. Can you find that point on that line? Where would that be? 

Kim  07:46

Yeah, but I kind of want... Like, I'm just like guessing. Like, why am I guessing? I need to go find it. That's silly.

Pam  07:54

Like, can you? If I'm looking at the line, and I literally took my cursor and I'm putting it on the line where x equals 10. You know, on the line. And it's sure enough, it's (10, 3). There it is. So, that's... There's one of the ones that you found. We could go find the other one. So, we can kind of have this conversation about how there's a lot of possibilities. In fact, if we let this line be, there's sort of an infinite set of possibilities. Now, if we round it off for money, then not so much. But there's a lot. A lot a lot of possibilities about what those sandwiches and cookies could cost. Cool. Alright, Kim, that was our first problem. Second problem. What if you went in and this time you bought 4 sandwiches and 5... Oh, golly, golly, golly. So, on my paper, I have written... Well, actually, on the board, I would just have the Desmos graph. 5x plus 4y equals 62. 

Kim  08:43

Mmhm.

Pam  08:44

On my paper, I wrote 5s plus 4c equals 62.

Kim  08:47

Mmhm.

Pam  08:48

For the second problem, I wrote 4s plus... Did I say 5 cookies? 

Kim  08:52

No.

Pam  08:52

4 sandwiches plus 5 cookies. And I mean, we just switched, so it should also be $62.00, right? Nope. Nope. Today it was $55.00. One less sandwich, one more cookie, and it costs less. Does that tell you anything? I'm just kind of curious. $55.00, yeah.

Kim  09:08

Say that again? It's 1 less sandwich and 1 more cookie. Oh, I wrote $62.00 again. It's how much? $55.00 1 less sandwich and 1 more cookie, and it's less. So, the sandwiches are more expensive. 

Pam  09:21

How do you know that? 

Kim  09:24

Because I have the same amount of things and it costs less. Like, when I added 1 more cookie and had 1 less sandwich, the price went down $7.00.

Pam  09:35

Yeah, so that sandwiches. Another way to look at it is to go from the one we just said. If you add a sandwich and take away a cookie, it went up by a lot. So, that's kind of interesting. Okay, cool. Do you think... What do you think that... How do I say this? Could we find some other possibilities for the price of sandwiches and the price of cookies? 

Kim  09:56

Yeah, you mean like on using Desmos. Like, you were to... I mean, I could come up with some random ones. 

Pam  10:02

Yeah.

Kim  10:03

You like me to?

Pam  10:05

Or... Yes. So, I might like look for one. Do you have one? Well, I have a fun one. What if the sandwiches were free? 

Kim  10:14

Okay.

Pam  10:15

So, 4 sandwiches are free. 5 cookies cost $55.

Kim  10:19

They're $11.00 bucks. 

Pam  10:20

Bam. So there's a... And where would we expect to see that? So, let's go ahead and type it now into Desmos. So, here's a different...

Kim  10:26

Do I need my original line? (unclear)

Pam  10:27

Yes, please keep your original one. Thank you. Great question. So,

Kim  10:30

You're wondering if I... Oh, there. I did it. Okay, good for me. 

Pam  10:31

what did you type in?

Kim  10:35

I haven't typed it in yet. I was excited that I found number two. Clearly, I do not do anything with Desmos.

Pam  10:43

I love it. 

Kim  10:43

Thank goodness my kids just...

Pam  10:46

They just come home and teach you all the Desmos. 

Kim  10:48

They do. They don't pay for that. Okay.

Pam  10:50

Alright.

Kim  10:50

Alright. New line. Green line.

Pam  10:52

Yep. Okay. And what did you type in?

Kim  10:55

4x plus 5y equals 55.

Pam  10:58

Do you dare try to tell me what that means?

Kim  11:02

5... 4 sandwiches and 5 cookies is $55.00 bucks.

Pam  11:05

Okay. And if I could 4 sandwiches times the price of the sandwiches plus (unclear)...

Kim  11:11

Oh yeah.

Pam  11:11

...times the price of the cookies is $55.00. Okay. And so, now that I'm assuming it graphed. And what do you see? Can you just describe for everybody what you see?

Kim  11:19

The line is shallower than the previous line. 

Pam  11:25

Okay. 

Kim  11:25

Like it's less slope.

Pam  11:27

Can you find that that cookie price when we said the sandwiches were free and the cookie was $11? 

Kim  11:34

Um.

Pam  11:36

Yeah. Maybe we should have put s and c. 

Kim  11:38

No, it's okay. So, I have (0, 11). I'm hovering over (0, 11). That's $0.00 cost for sandwiches and $11.00 for cookies. 

Pam  11:47

Nice, nice. And there's the intercept right there. And we could go look at the other intercept. It looks like it's not even. I've got (13.75, 0). So, I think that means, if the cookies were free and the sandwiches would cost $13.75 each. And then there's all sorts of possibilities in between.

Kim  12:07

Mmhm. Oh, I scrolled out enough, and then there's more to see. Haha. I was zoomed in too much still. So, the intersect. 

Pam  12:15

Tell me your window now because I kind of want you to stay in the window that I'm in. 

Kim  12:18

I'm in -25 to 30 on that x-axis. 

Pam  12:24

Oh, you moved around. I think that'll still work. Maybe don't. Maybe don't move around anymore. 

Kim  12:28

Okay. I was pretty zoomed in. 

Pam  12:30

Okay, okay.

Kim  12:31

I didn't see that they intersected.

Pam  12:34

Intersected. So, they do intersect. The two. And what's the intersection point?

Kim  12:39

(10, 3). 

Pam  12:40

And what do you think that means that (10, 3)

Kim  12:47

At $10.00 for the sandwich and $3.00 for the cookie. I think. It's $55.00 bucks. That's... Yeah, that's $55.00 bucks.

Pam  12:59

Okay.

Kim  12:59

But it's also when there's... What does that mean in the other line?

Pam  13:06

Yeah, what does that mean? 

Kim  13:06

$10.00 for the sandwiches and $4.00 for the cookies. No, 5 sandwiches and 4 cookies and 4 sandwiches and 5 cookies? That's where they are equivalent. Or they intersect.

Pam  13:19

What does it mean that those two... Like, one day I went to the cafe and I got 5 sandwiches and 4 cookies, and another day I went to the cafe and I got 4 sandwiches and 5 cookies. And they intersect.

Kim  13:29

They cost the same.

Pam  13:31

So, you think we're at the same cafe?

Kim  13:34

No, because one was $62.00 and one was $55.00, right? 

Pam  13:38

Yeah, but we bought different stuff.

Kim  13:42

Oh, yeah, we did. Haha.

Pam  13:43

What does the point (10, 3) mean? Like, that's the only place these two lines intersect, right?

Kim  13:49

Mmhm.

Pam  13:51

And I think you said for the second day we went in you said... Okay, so I bought 4 sandwiches at $10.00 each, and 5 cookies at $3.00 each, and it costs $55.00. And you can see that point there is (10, 3).

Kim  14:03

Mmhm.

Pam  14:04

But it's also on the other line. If I bought 5 sandwiches at $10.00 each, and 4 cookies at $3.00 each, it cost $62.00.

Kim  14:13

I don't think I understand what you're asking. 

Pam  14:14

Oh, maybe not. Sorry. I'm wondering what that point means for our scenario. And maybe you've already said it. When we did the first line, we had lots of different possibilities.

Kim  14:26

Sure.

Pam  14:27

That the sandwiches and cookies (unclear) costs. When we did the second line, we had other possibilities (unclear).

Kim  14:31

(unclear). So, this point, when I had the first cafe, it was $10.00 for the sandwiches and $3.00 for the cookies. And that's that point, (10, 3). So, in this scenario, it would be the $10.00 sandwich and the $3.00 cookie would be $55.00. Is that what you're asking?

Pam  14:49

I think so.

Kim  14:50

That's the price of the sandwiches and cookies when I...

Pam  14:54

On both days.

Kim  14:55

Yes.

Pam  14:55

Yeah, that's what I was trying to get at. 

Kim  14:57

Oh, sorry. I'm sorry.

Pam  14:59

So, we'll belabor it a little bit.

Kim  15:00

Didn't I say it was the price? 

Pam  15:01

Probably. 

Kim  15:02

Oh, okay.

Pam  15:03

Yeah, sorry.

Kim  15:03

Sorry.

Pam  15:04

Okay.

Kim  15:04

It's okay.

Pam  15:05

So, then you go back another day, and this day you bought 6 sandwiches.

Kim  15:09

6 sandwiches.

Pam  15:10

And 3 cookies. And today your bill was $69.00. And I'm curious, how that... Do you think you're in the same cafe? 

Kim  15:18

Oh, that's a good question. 

Pam  15:18

And what are you thinking to figure that out? 

Kim  15:19

I think that if the sandwiches were $10.00 bucks, that would be $60.00 bucks. And if the cookies were $3.00 bucks, that would be $9.00 bucks. And so, yes, I'm at the same cafe. 

Pam  15:23

Oh, so you tried it. You're like, "Let's stick it in there." What would it look like... Nice. What would it look like if you graph that line that represents 6 sandwiches and 3 cookies costing $69.00. Predict first.

Kim  15:44

You want me to predict first?

Pam  15:45

I do, yeah. If you don't mind. And, listeners, at home predicting what might that line look like.

Kim  15:50

So, I think the line where the second day the line was shallower of a slope. This time it's going to be a steeper slope.

Pam  16:00

Oh, interesting. Well, let's see if it is.

Kim  16:04

Find the third one. 6x plus 3y. 

Pam  16:10

Equals $69.00

Kim  16:13

Equals $69.00 Indeed.

Pam  16:15

Bam. And when you said "indeed", it's steeper. Is there anything else that pops out at you? 

Kim  16:21

The... I don't know the right word. The space between them. The steepness is...

Pam  16:29

Kind of mirrored?

Kim  16:30

Yeah, yeah, yeah.

Pam  16:31

Okay, alright, (unclear). 

Kim  16:32

Same change of slope.

Pam  16:33

Same symmetry going on. Okay. Or some symmetry going on. Anything else pop out at you like where that point (10, 3) was?

Kim  16:42

It's at 0. There's a... Oh, what about it? It's (10, 3). Yes.

Pam  16:46

Do all three lines now intersect in (10, 3)? 

Kim  16:49

Yep. 

Pam  16:49

So, we're in the same cafe. You already kind of said that numerically, but we can also kind of like. There's lots of...

Kim  16:55

I love sandwiches at this place apparently. I go three days in a row, but they're the same price, so that's exciting.

Pam  17:01

So, it's interesting that if I just said, Kim, you bought 6 sandwiches and 3 cookies, and they cost $69.00, could you have found lots of different combinations? Yes. But as we look at these different days at the cafe, you're getting different. There's lots of different possibilities for each one of them. But as we graph them, they're all intersecting in the same (10, 3), which means the price of the cookies and sandwiches has not changed.

Kim  17:22

Right.

Pam  17:23

Let's do one more. How about 3 sandwiches and 6 cookies. And that costs $48.00

Kim  17:29

Okay.

Pam  17:30

Cool. 

Kim  17:30

That makes sense.

Pam  17:31

Alright. And why does it make... What makes sense to you? 

Kim  17:34

Well, I'm thinking that the sandwiches are $10.00 bucks, so that's $30.00. 

Pam  17:37

Okay.

Kim  17:39

And then the cookies are $3.00 bucks, so that's $18.00.

Pam  17:43

Sure enough, yeah.

Kim  17:44

And that's $48.00.

Pam  17:46

Cool. And what might that line, if you were to predict that line, what might it look like?

Kim  17:50

It's going to be the most shallow. Why do you say that? What

Pam  17:53

is it about the things? You've kind of said "shallow" and "steep", but we haven't really talked about why? What about them?

Kim  18:00

When the price of the sandwich goes down... Man, how do I say this? The price of this... The number of sandwiches that I buy, goes down, it really reduces the total price more than if I were to buy 1 less cookie. Like... 

Pam  18:21

And that's... Yeah, sorry. Go ahead.

Kim  18:22

Yeah. I was just going to say because the sandwiches are more expensive when I buy 1 less of them, my total bill goes down more drastically.

Pam  18:28

Yeah.

Kim  18:28

Than if I were to just buy 1 less cookie at the time.

Pam  18:32

Yeah. And at some point you said something about a 7. 

Kim  18:37

Did I?

Pam  18:37

Remember when that was? Yeah, I wish I would have hung on to it better. But... 

Kim  18:41

Oh, the change in price is $7.00 because sandwiches are $10.00 bucks, and the cookies are $3.00, so the change overall is $7.00 bucks. 

Pam  18:52

So, like if you look between the original price of $62.00 and the next day's price of $55.00, there's a $7.00. And then from that original price of $62.00 to the third day price of $69.00, there's a change of $7.00. So, when we went up by a sandwich and down by a cookie, or vice versa, there was this change in 7 that was happening.

Kim  19:10

Right.

Pam  19:11

Which may or may not tell us the 10 and the 3, but it's kind of interesting that we kind of have that change in 7 going on. Cool. And did you graph 3x plus 6y equals 48? That fourth one? 

Kim  19:20

No, but I'm doing it right now.

Pam  19:21

Whoo!

Kim  19:22

Slow, but I'm there. Okay.

Pam  19:24

You're doing great. And does it intersect at (3, 10)? So...

Kim  19:31

Yeah.

Pam  19:31

Yep, got the same kind of thing going on? Cool. So, I could continue to keep giving you different sandwich numbers and different cookie numbers, and we could continue to see that because the cost of the sandwiches, the cookies aren't changing...

Kim  19:44

Yeah.

Pam  19:45

...that they are continuing to intersect at that intersection point. We could continue to look at different slopes changing. We could also look for some other patterns in the numbers that I give you. And one of those patterns that I'm just going to sort of mention because we're going to keep this podcast tight here today is I could have asked you 8 sandwiches and 10 cookies.

Kim  20:07

Mmhm. 

Pam  20:07

2 sandwiches and 7 cookies. And then I could have asked you for 10 sandwiches. So, maybe I'll just repeat the sandwiches in a row. I could have asked you for 8 sandwiches one day, given you the cookies and the cost. Then I could have asked you for 2 sandwiches, given you the cookies and the cost. Then I could have given you 10 sandwiches, given you the cookies and the cost. And a thing that I would have you notice, that I would ask questions about, is that you could actually add that 8 sandwich day and that 2 sandwich day to the 10 sandwich day, and notice some relationships happening. You could notice that you can actually add these orders together, and the cost of the sandwich and the cookies doesn't change. In other words, you can add the equations of lines together and the intersection point stays the same. And that was a huge stumbling block for me when I was learning systems of equations because they would say, "Here are two lines that are not the same in any way, but they intersect. Now, add them together, and you'll get a third line. And do stuff with it." And I was like, "I'm sorry, what?"

Kim  21:15

Yeah.

Pam  21:15

"You want me to add lines together and you're telling me that that has something to do with these first two lines?" It had made no sense to me that you could add equations of lines together, and that it would maintain the intersection point. You get a whole new line. It's not the same line.

Kim  21:32

Right.

Pam  21:32

It's just that... So, if you could look at kind of the spider that I have on my Desmos right now, I have lots of different equations of lines. They're all intersecting at that one (10, 3) because the price of the sandwich and the cookies didn't change, even if I made sure that I added different orders together and got different final prices. As long as that intersection point, or the price in the sandwich and the cookies, doesn't change, then I have a system of equations that's still equivalent. Which means they still have the same intersection point.  How cool is that? 

Kim  22:03

Yeah.  That's super cool.

Pam  22:05

Whoo!

Kim  22:05

Super cool.

Pam  22:07

Another thing that we could do if we had more time with this, and that I've done with with students in the past, is to kind of keep... At first, I kind of messed with the numbers not very systematically, but I could over time kind of get to where I had 2 sandwiches and 7 cookies, give you the price. Then I could down the sandwich and up the cookies. So, 2 sandwiches and 7 cookies, then I could down it and go 1 sandwich and 8 cookies. And then I could say, "Well, then let's down it one more time. What about 0 sandwiches and 9 cookies." And, bam, you've got the price for cookies.

Kim  22:41

Mmhm.

Pam  22:41

Right, because now you've got this sort of 0 thing happening. And I could have gone the other direction and walked back until I had 0 cookies and just sandwiches equaling that price. And then we could kind of solve for the sandwich price. So, ya'll, I would suggest that this is a fantastic task to do with students to have lots of ideas come together and really get kids thinking about systems of equations and making sense of it way before you do anything to have them think about graphing to find the solution, or using elimination, or using substitution. That this is the groundwork for systems. And it's hugely the groundwork for elimination. Bam! Alright, ya'll, stay tuned, and we'll do more systems of equations coming up soon. So, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. And keep spreading the word that Math is Figure-Out-Able!